from its location, and almost circular trajectories exist within the loops of the figure-eight. With a larger rod (right part of the figure), particles can escape from these loops and islands do not exist any more, resulting in better mixing. With a Poincaré section, the mixing quality of a flow can be analyzed by distinguishing between chaotic and elliptic regions. This is a crude measure of the mixing process, however, since the stretching properties cannot be inferred from this mapping method. Nevertheless, this technique is very useful for studying the mixing of periodic flows and can be extended to a 3-D domain. Fractal dimension Through a continual process of stretching and folding, much like in a "baker's map," tracers advected in chaotic flows will develop into complex fractals. The fractal dimension of a single contour will be between 1 and 2. Exponential growth ensures that the contour, in the limit of very long time integration, becomes fractal. Fractals composed of a single curve are infinitely long and when formed iteratively, have an exponential growth rate, just like an advected contour. The Koch Snowflake, for instance, grows at a rate of 4/3 per iteration. The figure below shows the fractal dimension of an advected contour as a function of time, measured in four different ways. A good method of measuring the fractal dimension of an advected contour is the uncertainty exponent. Evolution of tracer concentration fields in chaotic advection In fluid mixing, one often wishes to homogenize a species, that can be characterized by its concentration field q. Often, the species can be considered as a passive tracer that does not modify the flow. The species can be for example a dye to be mixed. The evolution of a concentration field obeys the advection-diffusion equation, also called convection–diffusion equation: Compared to the simple diffusion equation, the term proportional to the velocity field represents the effect of advection. When mixing a spot of tracer, the advection term dominates the evolution of the concentration field at the beginning of the mixing process. Chaotic advection transforms the spot into a bundle of thin filaments. The width of a dye filament decreases exponentially with time, until an equilibrium scale is reached, at which the effect of diffusion starts to be significant. This scale is called the Batchelor scale. It is defined as the square root of the ratio between the diffusion coefficient and the Lyapunov